International Journal of Computer Mathematics - Fast Iterative and Preconditioning Methods for Linear and Non-Linear Systems
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We present a new preconditioning method for a class of dense matrices, based on sparse approximation in a discrete wavelet basis. We first prove theoretical results that enable us to reduce the cost of an existing wavelet-based preconditioner and then incorporate those ideas into the design of our new preconditioner. We demonstrate the effectiveness of our methods with numerical examples drawn from one-dimensional physical problems and indicate how the method could be incorporated into a Kronecker product--based strategy for solving problems in higher dimensions. A notable feature of our new method is that it enables an optimal wavelet level to be chosen automatically, making it more robust than previous wavelet preconditioners that depend on the user choosing an appropriate level. Thus we are now closer to the ultimate goal of a black-box preconditioner for dense matrices.